7.4.2 Surface stress

The stress $\tau_{surf}$ at the ocean surface arises from two terms

 

$${\bf\tau}_{surf} {\bf\tau}_{fresh} + {\bf\tau}_{winds}.$$ = (7.55)

The first term is the usual wind stress contribution ${\bf \tau}_{wind}$. Unless MOM is coupled to a wave model which resolves the interactions between the sea surface and atmospheric winds, the surface wind stress is unresolved and so must be parameterized. As discussed in Section 4.3.3, MOM assumes that this parameterization takes the same aerodynamic form used for the bottom

 

$${\bf\tau}_{wind} \rho_{a} \, C_D^{wind} \, \vert{\bf u}^{wind}\vert\,{\bf u}^{wind}$$ = (7.56)

where C_D is a dimensionless drag coefficient and $\rho_{a}$ is the atmospheric density. This stress is formally identified with the friction terms evaluated at the ocean surface

 

$$\rho_{a} \, C_D^{wind} \, \vert{\bf u}^{wind}\vert\,{\bf u}^{wind} = \kappa_m {\bf u}_z - A \,(\nabla_{h}\eta) \cdot (\nabla_{h} {\bf u}).$$ (7.57)

In the linearized free surface as well as the rigid lid, the contribution from $A \,(\nabla_{h}\eta) \cdot (\nabla_{h} {\bf u})$, which arises from the curvature of the free surface, is absent. However, since the identification (7.57) is formal in the sense that no rigorous microscopic treatment of these friction terms is enabled in MOM, the differences are not important in practice. The stress ${\bf\tau}_{winds}$ is what MOM calls the ``surface momentum flux.'' In addition to the turbulent stress from the winds, fresh water entrained in atmospheric winds introduces momentum into the free surface ocean in the form given by equation (4.34)

$${\bf\tau}_{fresh} = q_w \, \rho_f \, {\bf u}^{wind},$$ (7.58)

where $\rho_f$ is the density of the fresh water. Equivalently, this stress can be thought of as the vertical advection of horizontal momentum across the ocean surface. In this case it should be proportional to the momentum of the fresh water outside the ocean. In general, it might be appropriate to consider the presence of an atmospheric model, a river model, or both, coupled to MOM. One may then wish to evaluate this stress from the zonal wind velocity and the zonal river current. Such detail, however, is not currently incorporated into MOM (although see Section 8.5 for a preliminary river runoff model). As a default, the simplest case

 

$$\tau_{fresh} \rho_{o} \, u(\eta) \, q_{w}$$ = (7.59)

is assumed. In actuality, there is always some difference in the wind speed and ocean current, and so some effective velocity may be more appropriate. Some consideration to this fact is given in the paper by Pacanowski (1987).

It is useful to further mention the importance of the stress from the fresh water term, especially in seas with a high fresh water flux. The implication of this term is that fresh water entrains also momentum if it enters the ocean with a horizontal velocity, which should be approximately the wind velocity. Thus, the above more complete boundary condition is necessary for an overall momentum conservation. An ocean with neither horizontal shear nor horizontal pressure gradients can be used as a simple test case to see the effect of the modified momentum flux boundary condition. It can be checked easily that the total momentum is constant if water is added at zero wind speed. If the wind speed is equal to the ocean surface velocity, there acts the usual wind stress but additionally the total momentum grows proportionally to the increasing surface height due to the input of volume through the fresh water. In summary, the complete free surface dynamic boundary condition takes the form

$$q_{w}\, {\bf u}_{h} + \kappa_m {\bf u}_z - A \,(\nabla_{h}\eta) \cdot (\nabla_{h} {\bf u}) \frac{\rho_{a}}{\rho }\, C_D^{wind} \, \vert{\bf u}^{wind}\vert\,{\bf u}^{wind} + q_{w}\, {\bf u}^{wind} \qquad z=\eta(\lambda,\phi,t).$$ =

The left hand side is the vertical momentum flux in the ocean, and the right hand side describes the vertical turbulent momentum flux in the atmosphere-ocean boundary layer. Note that for the fresh water volume flux the mass conserving form (see Section 4.4.3) has been used, although MOM strictly conserves only volume.